Base field 4.4.17600.1
Generator \(w\), with minimal polynomial \(x^{4} - 14x^{2} + 44\); narrow class number \(2\) and class number \(1\).
Form
| Weight: | $[2, 2, 2, 2]$ |
| Level: | $[19,19,-\frac{1}{2}w^{3} + w^{2} + 4w - 9]$ |
| Dimension: | $18$ |
| CM: | no |
| Base change: | no |
| Newspace dimension: | $34$ |
Hecke eigenvalues ($q$-expansion)
The Hecke eigenvalue field is $\Q(e)$ where $e$ is a root of the defining polynomial:
| \(x^{18} - 40x^{16} + 616x^{14} - 4678x^{12} + 18495x^{10} - 36933x^{8} + 34206x^{6} - 15148x^{4} + 3066x^{2} - 225\) |
Show full eigenvalues Hide large eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| 4 | $[4, 2, -\frac{1}{2}w^{3} + w^{2} + 4w - 8]$ | $\phantom{-}e$ |
| 11 | $[11, 11, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 4w + 11]$ | $\phantom{-}\frac{15461}{454128}e^{16} - \frac{155189}{113532}e^{14} + \frac{9621143}{454128}e^{12} - \frac{18463901}{113532}e^{10} + \frac{24758759}{37844}e^{8} - \frac{202726725}{151376}e^{6} + \frac{95258835}{75688}e^{4} - \frac{219572465}{454128}e^{2} + \frac{4435105}{75688}$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} - w + 2]$ | $\phantom{-}\frac{3661769}{13623840}e^{17} - \frac{28992919}{2724768}e^{15} + \frac{2195969249}{13623840}e^{13} - \frac{16225458137}{13623840}e^{11} + \frac{2035065799}{454128}e^{9} - \frac{36751464769}{4541280}e^{7} + \frac{26899602733}{4541280}e^{5} - \frac{23948006177}{13623840}e^{3} + \frac{778354223}{4541280}e$ |
| 11 | $[11, 11, -\frac{1}{2}w^{2} + w + 2]$ | $\phantom{-}\frac{477311}{3027520}e^{17} - \frac{3799543}{605504}e^{15} + \frac{290177671}{3027520}e^{13} - \frac{2172841533}{3027520}e^{11} + \frac{836912061}{302752}e^{9} - \frac{15828879173}{3027520}e^{7} + \frac{12866736571}{3027520}e^{5} - \frac{4377783723}{3027520}e^{3} + \frac{499061661}{3027520}e$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} - \frac{1}{2}w^{2} - 4w + 5]$ | $\phantom{-}\frac{3752327}{27247680}e^{17} - \frac{29615179}{5449536}e^{15} + \frac{2232671447}{27247680}e^{13} - \frac{16380784601}{27247680}e^{11} + \frac{2030651419}{908256}e^{9} - \frac{35870410687}{9082560}e^{7} + \frac{25077314269}{9082560}e^{5} - \frac{22342061171}{27247680}e^{3} + \frac{869490059}{9082560}e$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} + w^{2} - 4w - 9]$ | $-\frac{202039}{908256}e^{16} + \frac{7992901}{908256}e^{14} - \frac{120891847}{908256}e^{12} + \frac{890344615}{908256}e^{10} - \frac{554631709}{151376}e^{8} + \frac{1974315807}{302752}e^{6} - \frac{1394592907}{302752}e^{4} + \frac{1193529991}{908256}e^{2} - \frac{38537697}{302752}$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} - w^{2} - 4w + 9]$ | $-1$ |
| 19 | $[19, 19, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 5]$ | $\phantom{-}\frac{69227}{756880}e^{17} - \frac{69729}{18922}e^{15} + \frac{43368877}{756880}e^{13} - \frac{83391999}{189220}e^{11} + \frac{67082071}{37844}e^{9} - \frac{2739006161}{756880}e^{7} + \frac{1282546121}{378440}e^{5} - \frac{995037811}{756880}e^{3} + \frac{60329641}{378440}e$ |
| 25 | $[25, 5, \frac{1}{2}w^{2} - 1]$ | $-\frac{36227}{302752}e^{16} + \frac{1428039}{302752}e^{14} - \frac{21508323}{302752}e^{12} + \frac{157671297}{302752}e^{10} - \frac{293055893}{151376}e^{8} + \frac{1034860081}{302752}e^{6} - \frac{719902639}{302752}e^{4} + \frac{204690311}{302752}e^{2} - \frac{20773249}{302752}$ |
| 29 | $[29, 29, \frac{1}{2}w^{3} - 4w - 1]$ | $\phantom{-}\frac{352133}{851490}e^{17} - \frac{22348559}{1362384}e^{15} + \frac{424457521}{1702980}e^{13} - \frac{12605731807}{6811920}e^{11} + \frac{1594543943}{227064}e^{9} - \frac{1830448559}{141915}e^{7} + \frac{22289110283}{2270640}e^{5} - \frac{10393865411}{3405960}e^{3} + \frac{702930283}{2270640}e$ |
| 29 | $[29, 29, -\frac{1}{2}w^{3} + 4w - 1]$ | $-\frac{1016779}{13623840}e^{17} + \frac{8150561}{2724768}e^{15} - \frac{629105539}{13623840}e^{13} + \frac{4790005927}{13623840}e^{11} - \frac{632684177}{454128}e^{9} + \frac{12623011379}{4541280}e^{7} - \frac{11457403763}{4541280}e^{5} + \frac{13501738507}{13623840}e^{3} - \frac{562343353}{4541280}e$ |
| 31 | $[31, 31, -w - 1]$ | $\phantom{-}\frac{627245}{1816512}e^{17} - \frac{24808061}{1816512}e^{15} + \frac{375343445}{1816512}e^{13} - \frac{2768997707}{1816512}e^{11} + \frac{1732528469}{302752}e^{9} - \frac{6230141333}{605504}e^{7} + \frac{4504878367}{605504}e^{5} - \frac{3884156921}{1816512}e^{3} + \frac{116705185}{605504}e$ |
| 31 | $[31, 31, w - 1]$ | $-\frac{1088947}{6811920}e^{17} + \frac{8518583}{1362384}e^{15} - \frac{632956447}{6811920}e^{13} + \frac{4528540201}{6811920}e^{11} - \frac{534671555}{227064}e^{9} + \frac{8420521547}{2270640}e^{7} - \frac{3955347329}{2270640}e^{5} + \frac{746986051}{6811920}e^{3} + \frac{58356581}{2270640}e$ |
| 41 | $[41, 41, \frac{1}{2}w^{3} + \frac{1}{2}w^{2} - 4w - 2]$ | $\phantom{-}\frac{175447}{1513760}e^{17} - \frac{1406861}{302752}e^{15} + \frac{108624687}{1513760}e^{13} - \frac{827270331}{1513760}e^{11} + \frac{327843959}{151376}e^{9} - \frac{6540352861}{1513760}e^{7} + \frac{5932147317}{1513760}e^{5} - \frac{2325344551}{1513760}e^{3} + \frac{294471647}{1513760}e$ |
| 41 | $[41, 41, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 4w - 2]$ | $\phantom{-}\frac{886417}{3405960}e^{17} - \frac{870457}{85149}e^{15} + \frac{520755667}{3405960}e^{13} - \frac{470592377}{425745}e^{11} + \frac{226614553}{56766}e^{9} - \frac{7479204257}{1135320}e^{7} + \frac{2112043597}{567660}e^{5} - \frac{2285107261}{3405960}e^{3} + \frac{20413307}{567660}e$ |
| 49 | $[49, 7, -\frac{1}{2}w^{3} + \frac{1}{2}w^{2} + 5w - 7]$ | $\phantom{-}\frac{1924829}{6811920}e^{17} - \frac{15278245}{1362384}e^{15} + \frac{1162513709}{6811920}e^{13} - \frac{8663228447}{6811920}e^{11} + \frac{1104598117}{227064}e^{9} - \frac{20639506309}{2270640}e^{7} + \frac{16336338283}{2270640}e^{5} - \frac{15946835417}{6811920}e^{3} + \frac{567525653}{2270640}e$ |
| 49 | $[49, 7, \frac{3}{2}w^{3} - \frac{7}{2}w^{2} - 13w + 29]$ | $\phantom{-}\frac{152513}{756880}e^{17} - \frac{1202477}{151376}e^{15} + \frac{90488023}{756880}e^{13} - \frac{661600349}{756880}e^{11} + \frac{244275157}{75688}e^{9} - \frac{4237673579}{756880}e^{7} + \frac{2781725873}{756880}e^{5} - \frac{665140279}{756880}e^{3} + \frac{38863093}{756880}e$ |
| 59 | $[59, 59, -\frac{1}{2}w^{3} - \frac{3}{2}w^{2} + 3w + 10]$ | $-\frac{54959}{605504}e^{16} + \frac{2148295}{605504}e^{14} - \frac{31952879}{605504}e^{12} + \frac{229682057}{605504}e^{10} - \frac{412381749}{302752}e^{8} + \frac{1351763381}{605504}e^{6} - \frac{750486455}{605504}e^{4} + \frac{118500283}{605504}e^{2} + \frac{2369471}{605504}$ |
| 59 | $[59, 59, \frac{1}{2}w^{3} - \frac{3}{2}w^{2} - 3w + 10]$ | $-\frac{50835}{302752}e^{16} + \frac{2008461}{302752}e^{14} - \frac{30354611}{302752}e^{12} + \frac{223739079}{302752}e^{10} - \frac{419914063}{151376}e^{8} + \frac{1513510481}{302752}e^{6} - \frac{1110631873}{302752}e^{4} + \frac{344604907}{302752}e^{2} - \frac{38356171}{302752}$ |
| 61 | $[61, 61, \frac{1}{2}w^{2} + w - 6]$ | $-\frac{117403}{605504}e^{16} + \frac{4621491}{605504}e^{14} - \frac{69466683}{605504}e^{12} + \frac{507698429}{605504}e^{10} - \frac{938725913}{302752}e^{8} + \frac{3278214489}{605504}e^{6} - \frac{2207121571}{605504}e^{4} + \frac{580092855}{605504}e^{2} - \frac{50431701}{605504}$ |
Atkin-Lehner eigenvalues
| Norm | Prime | Eigenvalue |
|---|---|---|
| $19$ | $[19,19,-\frac{1}{2}w^{3} + w^{2} + 4w - 9]$ | $1$ |